Chebyshev's Inequality for Nonnegative Measurable Functions

Chebyshev's Inequality for Nonnegative Measurable Functions

Theorem 1 (Chebyshev's Inequality for Nonnegative Measurable Functions): Let $(X, \mathcal A, \mu)$ be a measure space and let $f$ be a nonnegative measurable function defined on $X$. Then for all $\lambda \in \mathbb{R}$, $\lambda > 0$ we have that $\displaystyle{\mu \{ x \in X : f(x) \geq \lambda \} \leq \frac{1}{\lambda} \int_X f(x) \: d \mu}$.
  • Proof: Define the set $X_{\lambda}$ by:
(1)
\begin{align} \quad X_{\lambda} = \{ x \in X : f(x) \geq \lambda \} \end{align}
  • Consider the function $\varphi = \lambda \chi_{X_{\lambda}}$. Then $\varphi$ is a simple function. Moreover, we have that $0 \leq \varphi \leq f$. Therefore:
(2)
\begin{align} \quad \lambda \mu (X_{\lambda}) = \int_X \varphi(x) \: d \mu \leq \int_X f(x) \: d \mu \end{align}
  • Since $\lambda > 0$ we divide both sides by $\lambda$ to get:
(3)
\begin{align} \quad \{ x \in X : f(x) \geq \lambda \} \leq \frac{1}{\lambda} \int_X f(x) \: d \mu \quad \blacksquare \end{align}
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